Modeling a twisty tree: How do I twist a beveled curve?
You're actually super close to getting what you want. The default circle shape of bevels just makes it really hard to see. The reason for this is because you're trying to twist a smooth, cylindrical object. Twisting a smooth cylinder results in another smooth cylinder, up until the point where you twist more than the geometry allows, which is what you see in the last gif.

I tried a similar setup but with an irregular shape and it works really well.

Here's my setup:



As you can see I used less handles, but that doesn't really matter.

I also animated the twist of each handle as I found it gave a nicer result. To find how much you need to twist each handle, think about how many twists you want overall, then divide that by the number of handles. 360 degrees per handle for 8 handles will give you a really twisted tree. The one in your image has more like 3 twists in the full length. In my case I did just over 2 full twists, so each handle was twisted 540 degrees more than the previous one. For your tree, try 90-120 degrees per handle and go from there.

When/if you animate these values, just be aware that because of the magnitude of the values, they will be off screen at first. 

Mine looked like this:



The final animation looks like this. The animations could be better, but the idea is there.